It won't affect your chances of getting a visa, but you will need to get a new passport in the long run. Why? Students are initially given a three month temporary visa by the German Embassy. Once they arrive in Germany, they are required to extend this visa at the Immigration office of their city.
Now here is where your ten months passport might be a problem. The Immigration office usually gives students a temporary residence permit of one year if they are able to show 8,640 euros in their blocked account. However, the number of years given by the Immigration office is also dependent on the number of months or years you have in your passport till it expires. For instance, if you have six months left in your passport, you will be given a residence permit of only six months instead of the one year even if you show the full 8,640 euros in your blocked account.
You mentioned that you are now about to start the visa application process. Considering that the whole visa application process takes a minimum of 3 months, that means that you will have around only seven months left in your passport if you are successful and able to make it to Germany. With only seven months left in your passport, the Immigration Office (Ausländerbehörde) will only give you a seven month temporary residence permit instead of one year even if you show the full money in your blocked account. So you end up losing in the long run.
You therefore have two options. You can either extend your passport in your home country or extend it once you arrive in Germany. The best will be to extend it at home before coming. I don't know about your country, but in most countries, you are only eligible to apply for a new passport if you have six months left to expiration. I'm sure you can find a way to extend it at home because you will have roughly seven months left in your passport once you are done with your visa processes, which is very little. If you are unable to extend it at home, you don't have to worry. You can still renew your passport at your consulate here in Germany. It costs around 140 euros and you usually get your new passport in 10 to 12 days.